Investment timing, debt structure, and financing constraints

ARTICLE IN PRESSJID: EOR [m5G;October 21, 2014;15:23]

European Journal of Operational Research 000 (2014) 1–14

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

Investment timing, debt structure, and financing constraints

Takashi Shibata a,∗, Michi Nishihara b,c

a Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-osawa, Hachioji, Tokyo 192-0397, Japanb Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japanc Swiss Finance Institute, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

a r t i c l e i n f o

Article history:

Received 27 June 2013

Accepted 1 September 2014

Available online xxx

Keywords:

Finance

Investment strategies

Real options

Debt structure

Debt issuance limit constraints

a b s t r a c t

We introduce debt issuance limit constraints along with market debt and bank debt to consider how finan-

cial frictions affect investment, financing, and debt structure strategies. Our model provides four important

results. First, a firm is more likely to issue market debt than bank debt when its debt issuance limit in-

creases. Second, investment strategies are nonmonotonic with respect to debt issuance limits. Third, debt

issuance limits distort the relationship between a firm’s equity value and investment strategy. Finally, debt

issuance limit constraints lead to debt holders experiencing low risk and low returns. That is, the more severe

the debt issuance limits are, the lower are the credit spreads and default probabilities. Our theoretical results

are consistent with stylized facts and empirical results.

© 2014 Elsevier B.V. All rights reserved.

1

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2 See, e.g., Stiglitz and Weiss (1981), Gale and Hellwig (1985), and Greenwald and

Stiglitz (1993) for static models of the investment decision under a financing constraint.

h

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. Introduction

In the frictionless financial markets of Modigliani and Miller

1958), investment and financing decisions are completely separa-

le. However, when we introduce financial frictions such as corpo-

ate taxes, bankruptcy costs, and financing constraints, there is no

onger separability between a firm’s investment and financing strate-

ies. Since their seminal study, the corporate finance literature has

ighlighted the role of financial frictions between investment and

nancing decisions.1

Brennan and Schwartz (1984) and Mauer and Triantis (1994) ex-

mine the interaction between investment and financing decisions in

ontingent claim models. These models have two major limitations.

irst, there are no financial frictions. Second, the firm is financed by a

ingle type of debt, not various debt structures. The drawback of treat-

ng corporate debt as uniform is highlighted by the fact that different

ypes of debt instruments have quite different effects on investment

trategies (see Bolton & Freixas, 2000; Hackbarth, Hennessy, & Leland,

007; Rauh & Sufi, 2010).

Several recent studies have already begun the task of incorporat-

ng either financing frictions or various debt structures separately into

nvestment timing decision (real options) models. Boyle and Guthrie

2003), Hirth and Uhrig-Homburg (2010), and Nishihara and Shibata

∗ Corresponding author. Tel.: +81 42 677 2310; fax: +81 42 677 2298.

E-mail addresses: [email protected] (T. Shibata), [email protected]

M. Nishihara).1 A partial list includes Fazzari, Hubbard, and Petersen (1988), Hoshi, Kashyap, and

charfstein (1991), Kaplan and Zingales (1997), Hennessy and Whited (2007), and

ivdan, Sapriza, and Zhang (2009).

N

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ttp://dx.doi.org/10.1016/j.ejor.2014.09.011

377-2217/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: T. Shibata, M. Nishihara, Investment timing

Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.09.011

2013) examine investment timing decisions under internal financing

onstraints. Nishihara and Shibata (2010) and Shibata and Nishihara

2012) investigate investment timing strategies under debt financ-

ng constraints.2 An interesting result among these earlier papers

s that investment strategies are nonmonotonic with respect to fi-

ancial frictions.3 Alternatively, most models consider a firm with

nly market debt (a single type of debt), assuming that dispersion

f creditors prevents debt reorganization during financial distress. In

ractice, a leveraged firm in financial distress can try to restructure

ts outstanding debt into a more affordable form. A model allow-

ng for debt restructuring approximates a firm with bank debt (i.e.,

onmarket debt).4 Sundaresan and Wang (2007) derive investment

trategies under various debt structures by considering debt reorga-

ization strategies for a firm under financial distress. However, these

trategies are derived independently of financing frictions.

In this paper, we assume that a firm can issue two classes of debt:

ank and market debt. Following Bulow and Shoven (1978), Gertner

nd Scharfstein (1991), and Hackbarth et al. (2007), the only differ-

nce between bank and market debt is the bankruptcy procedure.5

3 See Boyle and Guthrie (2003), Hirth and Uhrig-Homburg (2010), and Shibata and

ishihara (2012) for theoretical analyses. See Cleary, Povel, and Raith (2007) for an

mpirical analysis.4 A partial list of structural models with bank debt includes Mella-Barral and Per-

audin (1997), Fan and Sundaresan (2000), Broadie, Chernov, and Sundaresan (2007),

nd Hackbarth et al. (2007). None of these papers considers investment strategies.5 They assume that payments to market lenders cannot be changed outside the for-

al bankruptcy process and that the new owners can recapitalize optimally, although

osts are incurred.

, debt structure, and financing constraints, European Journal of

http://dx.doi.org/10.1016/j.ejor.2014.09.011

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2 T. Shibata, M. Nishihara / European Journal of Operational Research 000 (2014) 1–14


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Bank loan renegotiations are generally easier than bond restructur-

ing, as suggested in Lummer and McConnell (1989) and Gilson, John,

and Lang (1990). In addition, we assume that firms face issuance lim-

its for debt. It is widely observed that firms face debt issuance limits

in financing investment. Debt issuance limit constraints are typically

needed to rule out corporate default by mitigating risk shifting from

equity holders to debt holders (see Jensen & Meckling, 1976). Thus,

imposing debt issuance limit constraints in economic models can thus

capture an important feature of reality.

We introduce debt issuance limit constraints along with market

debt and bank debt to consider how financial frictions affect invest-

ment, financing, and debt structure strategies. Our model endoge-

nously determines investment timing, coupon payment level, and

debt structure (either bank or market debt issuance) in the presence

of financial frictions. To be more precise, our model is solved as fol-

lows. Given a debt structure, we derive the optimal investment and

coupon payment strategies under debt issuance limit constraints. We

then choose the optimal debt structure by comparing equity value

under bank debt with that under market debt.

Our model builds primarily on three papers: McDonald and Siegel

(1986), Sundaresan and Wang (2007), and Shibata and Nishihara

(2012).6 Our model becomes an all-equity financing model when the

firm cannot issue any type of debt (i.e., McDonald & Siegel, 1986).

Our model is a nonconstrained model under various debt structures

when the firm can issue bank and market debt without issuance con-

straints (i.e., Sundaresan & Wang, 2007). Our model becomes a con-

strained model under a market debt structure when a firm can issue

only market debt with an issuance bound constraint (i.e., Shibata

& Nishihara, 2012). Our model can be regarded as a natural exten-

sion of these three papers, and yields several additional important

implications.

Our model provides four important results. The first result is

that the type of debt that is issued at the time of investment de-

pends largely on three parameters (debt issuance limit, volatility, and

firm’s bargaining power during financial distress). The firm is more

likely to issue market debt than bank debt when the debt issuance

limit is increased.7 In practice, the firms with higher (lower) debt

issuance bounds are regarded as large/mature (small/young) corpo-

rations. Based on this definition, our results show that large/mature

(small/young) corporations are more likely to choose market (bank)

debt. Thus, this segmentation is broadly consistent with the stylized

facts. In addition, we consider the effects of volatility and bargaining

power. When volatility is sufficiently high, the firm prefers bank debt

financing irrespective of bargaining power. When volatility is high

and the firm’s bargaining power is low, the firm prefers bank debt

financing. The reason is that higher volatility is more likely to lead to

default, and a lower firm’s bargaining power is more likely to lead to

bank debt issuance. When volatility is low and the firm’s bargaining

power is high, the firm prefers market debt financing. This is because

lower volatility is less likely to result in default, and a higher firm’s

bargaining power causes less bank debt issuance. These findings fit

well with the empirical findings of Blackwell and Kidwell (1988) and

Denis and Mihov (2003).

The second result is that the investment strategies (one of two

solutions) are nonmonotonic with respect to the debt issuance con-

straints, while the coupon payments (the other solution) are mono-

tonic with respect to the debt issuance constraints. Given a debt struc-

ture, the investment thresholds have a U-shaped relationship with

the debt issuance constraints. The U-shaped relationship under bank

debt financing is a new result, although the U-shaped relationship

under market debt financing has already been established in Shibata

6 We incorporate the investment decision into a financing decision model having

the choice of debt structure developed by Hackbarth et al. (2007).7 This result is similar to that of Hackbarth et al. (2007) who do not consider invest-

ment strategies.

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Please cite this article as: T. Shibata, M. Nishihara, Investment timin


nd Nishihara (2012). The nonmonotonicity between investment and

rictions is similar to that in previous related papers (see, e.g., Boyle

Guthrie, 2003; Cleary et al., 2007; Hirth & Uhrig-Homburg, 2010).

iven a debt structure, the coupon payments are monotonically in-

reasing with respect to the debt issuance constraints. These results

mply that the optimal investment thresholds of the constrained lev-

red firm are not in between those of the unconstrained levered firm

nd unlevered firm, while the optimal coupon payments of the con-

trained levered firm are in between those of the unconstrained lev-

red firm and unlevered firm. Thus, it is less costly to distort in-

estment thresholds than to distort coupon payments. Moreover, if

he optimal debt structure strategies are changed from bank debt to

arket debt by increasing the debt issuance bound, the investment

hresholds and coupon payments have a discontinuous curved re-

ationship with respect to the debt issuance limit. In particular, the

nvestment thresholds may have a discontinuous W-shaped relation-

hip with the debt issuance limit.

The third result is about the relationship between equity option

alues and its investment thresholds. Suppose, as a benchmark, there

s no debt issuance limit constraint. Then, if the firm prefers bank

ebt financing, the investment thresholds under bank debt financ-

ng are lower than those under market debt financing. We call this

“symmetric relationship.” This implies that having the opportunity

o choose the debt structure always hastens corporate investment

decreases the investment threshold) and increases its equity options

alue. Suppose, by contrast, the firm is financially constrained by its

ebt issuance limit. Then we show that there is not always a symmet-

ic relationship between them. To be more precise, the investment

hresholds under bank debt financing are not always lower than those

nder market debt financing, even when the firm prefers bank debt.

his implies that having the opportunity to choose a debt structure

oes not always hasten corporate investment although its equity op-

ion value is increased. Thus, we show that financing constraints cause

distortion to the symmetric relationship between investment and

ts equity value.

The final result is that debt issuance constraints create a low-risk

nd low-return outcome for debt holders. In our model, return and

isk for debt holders can be measured by the credit spreads and default

robabilities, respectively. We show that, the more severe the debt is-

uance bounds are, the lower are the credit spreads and default prob-

bilities. Thus, we shed light on the determinants of the debt issuance

imits for debt holders. Moreover, we show that the credit spreads

nd default probabilities for bank debt are higher than those for mar-

et debt. These imply that bank debt is high risk and high return

ompared with market debt. These results are consistent with the

mpirical results of Davydenko and Strebulaev (2007).

The remainder of the paper is organized as follows. Section 2 de-

cribes the model and derives the value functions. Section 3 provides

he solution of our model and considers its properties. Section 4 dis-

usses the model’s implications. Section 5 concludes.

. Model

In this section, we begin with a description of the model. We

hen provide the value functions for the firms financed by bank debt

nd by market debt. Then, we formulate our model as an investment

ptimization problem for the firms financed by bank debt and by

arket debt with issuance limit constraints. As a benchmark, we

erive the solutions to the two extreme cases in our model. One is

he solution for an unlevered (all-equity financed) firm when the

ebt issuance limit constraints are strict (i.e., the model developed

y McDonald & Siegel, 1986). The other is the solution for a firm

nanced by bank debt and market debt when the debt issuance limit

onstraints are immaterial (i.e., a model similar to that of Sundaresan

Wang, 2007).

g, debt structure, and financing constraints, European Journal of


T. Shibata, M. Nishihara / European Journal of Operational Research 000 (2014) 1–14 3


0time

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T1i T

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region a

0time

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x2i

region a

T2d

region b

T2i

Fig. 1. Difference between market debt and bank debt.

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.1. Setup

A firm possesses an option to invest in a single project at any

ime. If the investment option is exercised at time T i (superscript

i” indicates investment strategies), the firm pays a fixed cost I > 0

t time T i and receives an instantaneous cash flow Xt after time T i,

here Xt is given by the following geometric Brownian motion:

Xt = μXtdt + σXtdzt, X0 = x, (1)

here zt denotes a standard Brownian motion, and the growth rate

> 0 and volatility σ > 0 are constant parameters. For convergence,

e assume that r > μ > 0,8 where r is a constant risk-free interest

ate. It is assumed that the current state variable X0 = x is sufficiently

ow so that the investment is not undertaken immediately.

In this paper, we assume that the firm issues two classes of per-

etual debt: market debt with a promised coupon payment flow c1

subscript “1” indicates market debt) and bank debt with a promised

oupon payment flow c2 (subscript “2” indicates bank debt). Follow-

ng Bulow and Shoven (1978), Gertner and Scharfstein (1991), and

ackbarth et al. (2007), we assume that the only difference between

ank debt and market debt is the bankruptcy procedure. Now sup-

ose that the firm in financial distress tries to restructure the debt.

nder market debt financing, the coupon payments to the market

ender cannot be changed outside the formal bankruptcy process.

nder bank debt financing, the coupon payments to the bank lender

re reduced in the course of a costless private workout.

Given a debt structure j ( j ∈ {1, 2}), let us denote by T ij

and Tdj

he investment (indicated by superscript “i” ) and default (indicated

y superscript “d”) timings, respectively. Mathematically, the invest-

ent and default timings are defined as T ij= inf{s ≥ 0 | Xs = xi

j} and

dj

= inf{s ≥ T ij| Xs = xd

j}, where xi

jand xd

jdenote the associated in-

estment and default thresholds, respectively. In particular, xd1 and xd

2

epresent the formal bankruptcy threshold for market debt and the

egotiation (coupon reduction) threshold for bank debt, respectively.

ote that these defaults are defined only after issuing the debt at the

8 The assumption r > μ ensures that the value of the firm is finite. See Sundaresan

nd Wang (2007), Kort, Murto, and Pawlina (2010), and Shibata and Nishihara (2011)

n detail.

“



ime of investment in this model (i.e., the firm is an unlevered firm

efore investment).

Fig. 1 illustrates the difference between market debt and bank

ebt. On the one hand, the left-hand side panel demonstrates the

ase of market debt financing. Once Xt , starting at X0 = x, arrives at

he threshold xi1, where superscript “i” indicates the investment, the

rm exercises the investment by issuing the market debt. If Xt hits

he formal bankruptcy threshold xd1 after adopting the investment,

here superscript “d” indicates the default, the firm is liquidated

the firm stops the operation). Mathematically, the investment and

ormal bankruptcy timings are defined as T i1 = inf{t ≥ 0 | Xt ≥ xi

1} andd1 = inf{t ≥ T i

1 | Xt ≤ xd1}.

On the other hand, the right-hand side panel depicts the case of

ank debt financing. Once Xt , starting at X0 = x, arrives at the invest-

ent threshold xi2, the firm exercises the investment by issuing the

ank debt. The investment timing is defined as T i2 = inf{t ≥ 0 | Xt ≥

i2}. If Xt hits the coupon switching threshold xd

2 from the above point,

he normal coupon payment is changed to the reduced coupon pay-

ent. In addition, if Xt hits the same threshold xd2 from below, the

oupon payment is changed to the normal coupon payment. The tim-

ngs for coupon switching are defined by Td2 := inf{t ≥ T i

2 | Xt = xd2},

rom the normal coupon to the reduced coupons as well as from the

educed coupon to the normal coupon.9 Thus, the firm pays the nor-

al coupon in the region Xt ≥ xd2, while the firm pays the reduced

oupon in the region Xt < xd2.10 Because the reduced coupon payment

n the region Xt < xd2 depends on Xt at the equilibrium, we can avoid

iquidation under bank debt financing.

.2. Value functions of the firm financed by market debt

In this subsection, we derive the value functions after the market

ebt is issued at the time of investment.

Let us denote by Ea1(Xt, c1) the equity value at time t after issuing

he market debt at the time of investment, where the superscript

a” represents firm value after the market debt issuance. The value,

9 For notational simplicity, we do not distinguish Td2 from above and below.

10 See Dixit (1989) for the switching model.





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θ

2

Ea1(Xt, c1), is defined as

Ea1(Xt, c1) = sup

Td1

Et

[ ∫ Td1

t

e−r(u−t)(1 − τ)(Xu − c1)du

], (2)

where Et denotes the expectation operator at time t, and τ denotes

the tax rate. Note that Td1 is the formal bankruptcy time that the equity

holders optimize. As in standard arguments, Ea1(Xt, c1) is given by

a1(Xt, c1)

= maxxd

1

{�Xt − (1 − τ)

c1

r−

(�xd

1 − (1 − τ)c1

r

)(Xt

xd1

)γ }, (3)

where � := (1 − τ)/(r − μ) > 0 and γ := 1/2 − μ/σ 2 − ((μ/σ 2 −1/2)2 + 2r/σ 2)1/2 < 0. Then, the optimal threshold for formal

bankruptcy is obtained by

xd1(c1) = argmax

xd1

Ea1(Xt, c1) = κ−1

1 c1, (4)

where κ1 := (γ − 1)�r/(γ (1 − τ)) > 0. Note that xd1(c1) is given as a

linear function of c1 with limc1↓0 xd1(c1) = 0. The result is obtained by

Black and Cox (1976).

The debt value, Da1(Xt, c1), is given as

Da1(Xt, c1) = Et

[ ∫ Td1

t

e−r(u−t)c1du + e−r(Td1 −t)(1 − α)�xd

1(c1)

]

= c1

r+

((1 − α)�xd

1(c1)− c1

r

)(Xt

xd1(c1)

)γ

, (5)

where α ∈ (0, 1) denotes the proportional cost of formal bankruptcy.

Because limc1↓0 xd1(c1) = 0 in (4), we have limc1↓0 Da

1(Xt, c1) = 0. The

total firm value is defined by the sum of the equity and debt values,

Va1(Xt, c1) := Ea

1(Xt, c1)+ Da1(Xt, c1), i.e.,

Va1(Xt, c1) = �Xt + τ c1

r

(1 −

(Xt

xd1(c1)

)γ )− α�xd

1(c1)

(Xt

xd1(c1)

)γ

.

(6)

There are three components of the right-hand side in (6). The first,

second, and third terms represent the values of the unlevered firm, the

tax shield, and the formal bankruptcy cost, respectively. Obviously,

we have limc1↓0 Va1(Xt, c1) = �Xt .

2.3. Value functions for the firm financed by bank debt

This subsection provides the value functions after the bank debt

is issued at the time of investment.

When the bank debt is issued at the time of investment, there are

two types of regions: normal and renegotiation regions. On the one

hand, let us denote by Ea2(Xt, c2), Da

2(Xt, c2), and Va2(Xt, c2) the equity,

debt, and total firm values, respectively, in the normal region, where

the superscript “a” stands for the normal region after issuing the

bank debt. On the other hand, let us denote by Eb2(Xt, c2), Db

2(Xt, c2),

and Vb2 (Xt, c2) the equity, debt, and total firm values, respectively,

in the negotiation region during the period of financial distress. The

superscript “b” indicates the negotiation (coupon reduction). The nor-

mal and negotiation regions are divided by the negotiation (coupon

reduction) threshold xd2. That is, the region {Xt ≥ xd

2} is the normal

region, while the region {Xt < xd2} is the negotiation region. The firm

and bank negotiate and divide the surplus Vb2 (Xt, c2)− (1 − α)�Xt to

avoid formal bankruptcy. The division of the surplus between the firm

and the bank depends on their relative bargaining powers. Let us de-

note by η and 1 − η the bargaining powers of the firm and the bank,

respectively (η ∈ [0, 1]).

The equity and debt values in the normal region, Ea2(Xt, c2) and

Da2(Xt, c2), respectively, are given by

Ea2(Xt, c2) = sup

Td2 ≥t

Et

[ ∫ Td2

t

e−r(u−t)(1 − τ)(Xu − c2)du



+ e−r(Td2 −t)Eb

2(XTd2, c2)

], (7)

a2(Xt, c2) = Et

[ ∫ Td2

t

e−r(u−t)c2du + e−r(Td2 −t)Db

2(XTd2, c2)

]. (8)

ere, the coupon switching times are defined by Td2 = inf{t ≥ 0, Xt =

d2}. The equity and debt values in the negotiation region, Eb

2(Xt, c2)

nd Db2(Xt, c2), respectively, are given by

Eb2(Xt, c2) = Et

[ ∫ Td2

t

e−r(u−t)((1 − τ)Xu − s(Xu))du

+ e−r(Td2 −t)Ea

2(XTd2, c2)

], (9)

b2(Xt, c2) = Et

[ ∫ Td2

t

e−r(u−t)s(Xu)du + e−r(Td2 −t)Da

2(XTd2, c2)

],

(10)

here s(Xt) is the reduced coupon payment in the negotiation region.

he total firm value is defined by Vl2(Xt, c2) = El

2(Xt, c2)+ Dl2(Xt, c2)

l ∈ {a, b}).

We begin by solving the total firm values. By the standard variation

rinciple, we have the following differential equations:

rVa2(x, c2) = (1 − τ)x + τ c2 + μx

∂Va2(x, c2)

∂x

+ 1

2σ 2x2 ∂2Va

2(x, c2)

∂x2, Xt ≥ xd

2, (11)

Vb2 (x, c2) = (1 − τ)x + μx

∂Vb2 (x, c2)

∂x

+ 1

2σ 2x2 ∂2Vb

2 (x, c2)

∂x2, Xt < xd

2, (12)

ubject to the boundary conditions

a2

(xd

2, c2

) = Vb2

(xd

2, c2

),

∂Va2(x, c2)

∂x

∣∣∣∣x=xd

2

= ∂Vb2 (x, c2)

∂x

∣∣∣∣x=xd

2

.

hus, Va2(x, c2) and Vb

2 (x, c2) are obtained by

Va2(Xt, c2) = �Xt + τ c2

r

(1 − β

β − γ

(Xt

xd2

)γ ), Xt ≥ xd

2, (13)

b2 (Xt, c2) = �Xt − τ c2

r

γ

β − γ

(Xt

xd2

)β

, Xt < xd2, (14)

here β := 1/2 − μ/σ 2 + ((μ/σ 2 − 1/2)2 + 2r/σ 2)1/2 > 1. Note that

here is no bankruptcy cost term of Va2(Xt, c2) in (13), in contrast with

a1(Xt, c1) in (6).

We then derive the reduced coupon payment in the negotiation

egion. As in the derivation in Sundaresan and Wang (2007), the re-

uced coupon payment in the renegotiation region, s(Xt), is given by

(Xt) = (1 − αη)(1 − τ)Xt, Xt < xd2. (15)

he reduced coupon payment s(x) is a linear function of x with

imx↓0 s(x) = 0. The lower the EBIT x is, the lower is the coupon pay-

ent s(x). This means that the firm does not need to consider formal

ankruptcy once it is in the negotiation region.

Let θe(Xt, c2) denote the fraction of firm value Vb2 (Xt, c2) that the

rm receives from renegotiation. Using Nash bargaining, θe(Xt, c2) is

btained by solving

e(Xt, c2) = argmaxθ

(θVb

2 (Xt, c2))η

× ((1 − θ)Vb

2 (Xt, c2)− (1 − α)�Xt

)1−η

= η − η(1 − α)�Xt

Vb(Xt, c2).





T

D

w

E

E

T

∂

x

w

x

i

D

I

2

f

c

W

s

f

t

d

i

m

e

a

1

r

D

c

t

s

r

t

E

s

P

n

u

m

v

E

w

E

f

m

r

t

d

r

t

w

2

c

fi

l

(

E

f

v

x

w

2

T

n

s

T

E

w

E

f 1N 2N

11 Hackbarth et al. (2007) consider the mixed debt policy because they incorporate

the priority structures of bank and market debt. Note that we do not introduce them.12 This problem is equivalent to the investment problem for an all-equity financed

firm, which is the simple version of the seminal model by McDonald and Siegel (1986).13 This problem is similar to that in Sundaresan and Wang (2007).

hus, the sharing rules are

Eb2(Xt, c2) = ηVb

2 (Xt, c2)− η(1 − α)�Xt

= η

(α�Xt − τ c2

r

γ

β − γ

(Xt

xd2

)β), (16)

b2(Xt, c2) = (1 − η)Vb

2 (Xt, c2)+ η(1 − α)�Xt

= (1 − αη)�Xt − (1 − η)τ c2

r

γ

β − γ

(Xt

xd2

)β

, (17)

here Xt < xd2. Moreover, the equity value in the normal region,

a2(Xt, c2) in (7), is rewritten as

a2(Xt, c2) = max

xd2

{�Xt − (1 − τ)

c2

r

−((1 − αη)�xd

2 − c2

r

(1 − τ − τ

ηγ

β − γ

))(Xt

xd2

)γ }.

(18)

he optimal coupon switching threshold is chosen to satisfy

Ea2/∂xd

2 = ∂Eb2/∂xd

2, i.e.,

d2(c2) = κ−1

2 c2, (19)

here κ2 := (γ − 1)(1 − αη)�r/(γ (1 − τ(1 − η)) > 0. Note thatd2(c2) is a linear function of c2 with limc2↓0 xd

2(c2) = 0. The debt value

n the normal region, Da2(Xt, c2) in (8), is given by

a2(Xt, c2) = c2

r+

((1 − αη)�xd

2(c2)

− c2

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

))(Xt

xd2(c2)

)γ

. (20)

t is straightforward to obtain limc2↓0 Da2(Xt, c2) = 0.

.4. Investment problem for the constrained levered firm

In this subsection, we formulate the investment decision problem

or the firm financed by bank debt and by market debt with issuance

onstraints.

Suppose that the debt structure j is given ( j ∈ {1, 2}). Following

ong (2010) and Shibata and Nishihara (2012), the firm’s debt is-

uance limit constraint is assumed to be

Daj

(xi

j, cj

)I

≤ q, (21)

or some constant q ≥ 0. The inequality (21) means that the ratio of

he debt issuance amount to the investment cost is restricted by the

ebt issuance ratio q. In other words, the firm has access to debt

ssuance Daj(xi

j, cj) up to the amount qI ≥ 0. For example, the friction

ight arise because of the risk-shifting problem. Because issuing debt

ncourages risk shifting from equity holders to debt holders, investors

re reluctant to lend beyond a certain amount (see Jensen & Meckling,

976). Note that we have assumed q ∈ [0,+∞), not q ∈ [0, 1]. The

eason is that the levered firm has the possibility of issuing the amountaj(xi

j, cj), which is more than the amount I, to maximize its value. The

ase of Daj(xi

j, cj) > I may lead to the result q > 1, which implies that

he excess is distributed to equity holders as a dividend.

Let us denote by Eoj(x) the equity options value of the firm con-

trained by debt issuance limit ( j ∈ {1, 2}), where the superscript “o”

epresents the options value before investment. Given a debt struc-

ure j, the equity options value of the constrained firm is defined as

oj (x) := sup

T ij≥0,cj≥0

E0[e−rT i

j {Eaj (XT i

j, cj)− (

I − Daj (XT i

j, cj)

)}],

ubject to Daj(X

Tij, cj) ≤ qI. Using standard arguments in Dixit and

indyck (1994), we have E0[e−rTi

j ] = (x/xij)β where X0 = x < xi

j.



We formulate the investment decision problem of the firm fi-

anced by bank debt and market debt with issuance constraints. Let

s denote by Eo(x) the equity options value of the firm with opti-

al debt structure subject to issuance constraints. The equity options

alue Eo(x) is obtained by

o(x) = max {Eo1(x), Eo

2(x)}, (22)

here Eoj(x) is given by

oj (x) = max

xij≥0,cj≥0

(x

xij

)β

{Vaj

(xi

j, cj

) − I}, (23)

subject toDa

j

(xi

j, cj

)I

≤ q, j ∈ {1, 2}, (24)

or any x < min{xi1, xi

2}.

This problem is related to the issuance of either bank debt or

arket debt at the time of investment. The reason is that we can

emove the mixed issuance of bank debt and market debt (we show

hat this assumption is correct below).11

Before analyzing the optimal investment strategies for a firm with

ifferent debt structures subject to a debt issuance constraint, we first

eview two extreme cases briefly: the optimal investment problems of

he unlevered firm and the levered firm with different debt structures

ithout issuance constraints.

.5. Investment problem for the unlevered firm

In this subsection, we assume that q = 0, i.e., Daj(xi

j, 0) = 0 and

j = 0 for all j ( j ∈ {1, 2}).12

Let us denote by EoU(·) the equity options value of the unlevered

rm before investment, where the subscript “U” stands for the un-

evered firm. When q = 0, we have Eaj(xi

j, 0) = Va

j(xi

j, 0) = �xi

jfor all j

j ∈ {1, 2}). Thus, the value, EoU(x), turns out to be

oU(x) = max

xiU≥0

(x

xiU

)β

{�xiU − I}, (25)

or any x < xiU. The optimal investment strategy and the equity options

alue are

i∗U = β

β − 1

1

�I, Eo

U(x) =(

x

xi∗U

)β

(β − 1)−1I, (26)

here the superscript “∗” represents the optimum.

.6. Investment problem for the unconstrained levered firm

In this subsection, we assume that q is sufficiently large (q ↑ +∞).

hen, the debt issuance constraint becomes immaterial.13

Let us denote by EoN(x) the equity options value for the firm fi-

anced by bank debt and market debt without an issuance con-

traint, where the subscript “N” represents the nonconstrained firm.

he value, EoN(x), is formulated as

oN(x) = max {Eo

1N(x), Eo2N(x)}, (27)

here EojN

(x) is given by

ojN(x) = max

xijN

≥0,cjN≥0

(x

xijN

)β

{Vaj

(xi

jN, cjN

) − I}, (28)

or any x < min{xi , xi } ( j ∈ {1, 2}).





00

x

Eo (

equi

ty o

ptio

n va

lue)

EUo (x)

EUa(x)

EjNo (x)

Eja(x,c

jN)

xU∗

(β−1)−1I

xjN∗

Fig. 2. Equity option values and investment thresholds without financing constraints.

3

i

p

t

i

d

p

3

v

D

w

g

e

(

t

0

v

t

l

d

h

c

x

t

t

t

L

x

O

f

c

r

P

j

s

w

f

f

f

The solutions and equity option value for a given debt structure j

are14

xi∗jN = ψjx

i∗U , c∗

jN = κjψj

hj

xi∗U , Eo

jN(x) = ψ−βj

EoU(x), (29)

where

h1 :=(

1 − γ

(1 + α

1 − τ

τ

))−1/γ

≥ 1,

ψ1 :=(

1 + τ

1 − τ

1

h1

)−1

≤ 1,

h2 :=(

β

β − γ(1 − γ )

)−1/γ

≥ 1,

ψ2 :=(

1 + τ(1 − αη)

1 − τ(1 − η)

1

h2

)−1

≤ 1. (30)

We summarize these results obtained in (29) as follows.

Lemma 1. Suppose that the firm does not have a debt issuance constraint

(i.e., q is sufficiently large). The first property is that we have xi∗jN

≤ xi∗U and

EojN

(x) ≥ EoU(x) for any j ( j ∈ {1, 2}). The second property is that ψ1 < ψ2

implies that xi∗1N < xi∗

2N and Eo1N(x) > Eo

2N(x). The third property is that

EoU(xi∗

U ) = Eo1(x

i∗1N) = Eo

2(xi∗2N).

Lemma 1 is easily demonstrated by Fig. 2 which depicts the equity

option value with the state variable. The first property is the same

as that in the static model of Myers (1977). The second property

corresponds to the third property. From these two properties, we

have xi∗1N < xi∗

2N if and only if Eo1N(x) > Eo

2N(x). We call this property

symmetric relationship.

Before solving our problem, we consider the properties of the so-

lution and its value. We have the following two intuitive conjectures:

xi∗j ∈ [xi∗

jN, xi∗U ] for any q, xi∗

1 ≤ xi∗2 if and only if Eo

1(x) ≥ Eo2(x).

The first conjecture is implied by the fact that xi∗U = limq↓0 xi∗

j>

limq↑+∞ xi∗j

= xi∗jN

for any q ( j ∈ {1, 2}). The second one is implied by

the fact that xi∗1N ≤ xi∗

2N if and only if Eo1N(x) ≥ Eo

2N(x). However, inter-

estingly, these intuitive conjectures are not necessarily correct.

14 See Sundaresan and Wang (2007) and Shibata and Nishihara (2010) for a detailed

derivation.

f



. Model solution

In this section, we derive the solution to the problem formulated

n (22). Section 3.1 provides the solution to the investment decision

roblem for the constrained levered firm with a given debt struc-

ure. Section 3.2 examines the optimal debt structure by compar-

ng the equity options values for a firm with bank debt and market

ebt with issuance constraints. Then, we obtain the solution to the

roblem (22).

.1. Optimal investment strategies for a given debt structure

Given a debt structure j ( j ∈ {1, 2}), we define the constant critical

alue xpj

satisfying

aj

(x

pj, cj

(x

pj

)) = qI, (31)

here the superscript “p” represents the critical point and cj(x) is

iven as the optimal coupon payment of the nonconstrained lev-

red firm given a debt structure j,15 i.e., cj(x) := argmaxcjVa

j(x, cj) =

κj/hj)x as shown in Leland (1994). Then, Daj(x, cj(x))is a strictly mono-

onically increasing continuous function of x with limx↓0 Daj(x, cj(x)) =

and limx↑+∞ Daj(x, cj(x)) = +∞. Thus, there exists a unique critical

alue xpj

.

Suppose, for example, xi∗jN

> xpj

for any j ( j ∈ {1, 2}), implying that

he investment threshold of the nonconstrained levered firm is strictly

arger than the critical value. Then, the firm would prefer to issue more

ebt than the amount qI to maximize the equity value, while the debt

olders restrict the amount to less than qI. Thus, the firm is financially

onstrained by its debt issuance bound. Suppose that, instead, xi∗jN

≤pj

. Then, the firm can choose the investment threshold to maximize

he total firm value on the condition that the debt issuance is less than

he bound qI. Therefore, the firm is not constrained. We summarize

hese results as follows.

emma 2. Given a debt structure j ( j ∈ {1, 2}), there exists a uniquepj

satisfying (31). Then, if xi∗jN

> xpj

, the firm is financially constrained.

therwise, it is not.

Lemma 2 implies that, by comparing the magnitudes of xi∗jN

and xpj

or a given debt structure j, we recognize whether the debt issuance

onstraint is binding. Using these properties, we have the following

esults (the proof is given in Appendix A).

roposition 1. Suppose that xi∗jN

> xpj

≥ 0 for a given debt structure

(j ∈ {1, 2}). For xpj

= 0, we have xi∗j

= xi∗U and c∗

j= 0. For x

pj

> 0, the

olutions, xi∗j

and c∗j, are decided uniquely by two simultaneous equations:

fj1

(xi∗

j, c∗

j

)fj2

(xi∗

j, c∗

j

) −fj3

(xi∗

j, c∗

j

)fj4

(xi∗

j, c∗

j

) = 0, Daj (x

i∗j , c∗

j )− qI = 0, (32)

here fj1, fj2, fj3, and fj4 are given as

11 := (β − 1)�xi∗1

+(β

τ

r− (β − γ )

(κ1xi∗

1

c∗1

)γ (τ

r+ α�

κ1

))c∗

1 − βI, (33)

12 :=(

κ1xi∗1

c∗1

)γ

γ

(1

r− (1 − α)

�

κ1

)c∗

1, (34)

13 := τ

r−

(κ1xi∗

1

c∗1

)γ

(1 − γ )

(τ

r+ α�

κ1

), (35)

14 := 1

r−

(κ1xi∗

1

c∗

)γ

(1 − γ )

(1

r− (1 − α)�

κ1

), (36)

1

15 Note that we have cj(xi∗jN) = c∗

jN, where c∗

jNis defined by the second term of (29).





a

f

f

f

f

S

o

v

E

3

v

e

o

o

p

τi

(

E

E

t

o

a

n

E

a

v

O

l

i

w

F

w

c

p

p

w

fi

c

v

{

l

o

p

l

a

E

r

O

l

d

F

b

o

a

s

l

d

w

I

w

t

a

w

4

o

c

s

t

w

d

c

d

h

4

o

w

h

C

l

i

t

(

f

t

h

nd

21 := (β − 1)�xi∗2 + β

τ

r

(1 −

(κ2xi∗

2

c∗2

)γ )c∗

2 − βI, (37)

22 :=(

κ2xi∗2

c∗2

)γ

×γ

(1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

)− (1 − αη)

�

κ2

)c∗

2,

(38)

23 := τ

r

(1 −

(κ2xi∗

2

c∗2

)γ

(1 − γ )β

β − γ

), (39)

24 := 1

r−

(κ2xi∗

2

c∗2

)γ

(1 − γ )

×(

1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

)− (1 − αη)

�

κ2

). (40)

uppose, on the other hand, that xi∗jN

≤ xpj

for a given debt structure j. We

btain xi∗j

= xi∗jN

and c∗j

= c∗jN

. Substituting the solutions into the equity

alue in (23) yields

oj (x) =

(x

xi∗j

)β

{Vaj

(xi∗

j , c∗j

) − I}. (41)

.2. Optimal debt structure strategies

In the previous subsection, we considered the solutions and their

alues for a given debt structure j ( j ∈ {1, 2}). In this subsection, we

xamine the solutions to the problem (22). Recall that we cannot

btain the analytical solution of Eoj(x). By comparing the magnitudes

f Eo1(x)and Eo

2(x) for given parameters, we obtain the solutions to the

roblem defined by (22) numerically.

Suppose that the basic parameters are r = 0.09, μ = 0.01, I = 5,

= 0.15, α = 0.4, and x = 0.4.16 Whether bank or market debt is

ssued depends on the combination of the three key parameters: q

debt issuance limit), σ (volatility), and η (firm’s bargaining power).

The upper left-hand side panel of Fig. 3 depicts the regions ofo1(x) > Eo

2(x) in (η,σ ) space. Three lines indicate the boundaries ofo1(x) = Eo

2(x) for q = 1, q = 0.8, and q = 0.6. Under the basic parame-

ers, the boundary for q = 1 is the same as the one for q ↑ +∞ (i.e., the

ne for the nonconstrained levered firm). For the regions of smaller σnd larger η, we see Eo

1(x) > Eo2(x), i.e., the firm prefers market debt fi-

ancing. For the regions of larger σ and smaller η, in contrast, we haveo2(x) > Eo

1(x), i.e., the firm prefers bank debt financing. Importantly,

n increase in q enlarges the regions of Eo1(x) > Eo

2(x). The next obser-

ation characterizes the effects of debt issuance limit parameter q.

bservation 1. Consider the optimization problem for the constrained

evered firm defined by (22). First, as debt issuance limit constraint q

ncreases, the firm becomes more likely to issue market debt. Second,

hen volatility σ is greater, the firm is more likely to prefer bank debt.

inally, when the firm’s bargaining power η is stronger (in other words,

hen the bank’s bargaining power is weaker), the firm is more likely to

hoose market debt.

In the upper left-hand side panels of Fig. 3, we consider three

oints: Points A, B, and C. Point A is an arbitrary case where the firm

refers market debt financing for any q. Point B is an arbitrary case

here the firm prefers market debt financing for larger q, while the

rm prefers bank debt financing for smaller q. Point C is an arbitrary

ase where the firm prefers bank debt financing for any q.

The other three panels of Fig. 3 demonstrate the equity options

alues with respect to q. The parameters of the upper right-hand,

16 Under the basic parameters, x = 0.4 always satisfies x < min{xi∗j, xi∗

jN} for all j ( j ∈

1, 2}).

h

(



ower left-hand, and lower right-hand side panels correspond to those

f Points A, B, and C, respectively. First, in the upper right-hand side

anel, we confirm Eo1(x) ≥ Eo

2(x) for all regions of q. Second, in the

ower left-hand side panel, we see that Eo2(x) > Eo

1(x) for q < q̂ = 0.71

nd Eo1(x) ≥ Eo

2(x) for q ≥ q̂. Finally, in the lower right-hand side panel,o2(x) ≥ Eo

1(x) for all regions of q. Consequently, we have the following

esult.


evered firm defined by (22). For η ∈ (0, 1], there are three optimal cases,

epending on the parameters.

(i) There exists Eo2(x) ≥ Eo

1(x)for all regions of q. Then, the constrained

levered firm chooses bank debt issuance.

(ii) There exists a unique q̂ such that Eo2(x) ≥ Eo

1(x) for q ≤ q̂ and

Eo1(x) > Eo

2(x) for q > q̂. Then, the firm prefers bank debt issuance

for q ≤ q̂, while it prefers market debt issuance for q > q̂.

(iii) There exists Eo1(x) ≥ Eo

2(x) for all regions of q. Then, the firm

chooses market debt issuance.

or η = 0, on the other hand, there exists only one case. The firm chooses

ank debt issuance at the time of investment.

Observations 1 and 2 are most closely related to the stylized facts

n debt composition. According to the definition by Rajan (1992)

nd Hackbarth et al. (2007), the firms with larger q, larger η, and

maller σ (smaller q, smaller η, and larger σ ) best approximate

arge/mature (small/young) corporations. Then, based on the above

efinition, small/young firms are more likely to issue bank debt,

hereas large/mature firms are more likely to issue market debt.

n addition, these implied results in Observations 1 and 2 fit well

ith the empirical findings. Blackwell and Kidwell (1988) observe

hat small and risky firms sell debt privately. Denis and Mihov (2003)

nd Rauh and Sufi (2010) find that low-credit-quality firms (the firm

ith high volatility) are more likely to have bank debt.

. Model implications

In this section, we consider the more important implications of

ur model. Sections 4.1 and 4.2 investigate the effects of debt issuance

onstraints on the investment thresholds and coupon payments, re-

pectively. Section 4.3 shows that a symmetric relationship between

he investment thresholds and the corresponding values does not al-

ays exist. Section 4.4 considers the effects of having the choice of

ebt structure. Section 4.5 analyzes the effects of a debt issuance limit

onstraint on the credit spreads and default probabilities. Section 4.6

iscusses how the debt issuance amount is determined by debt

olders.

.1. Effects of constraints on investment thresholds

This subsection examines the effects of a debt issuance constraint

n the investment thresholds.

The upper two panels of Fig. 4 illustrate the investment thresholds

ith the debt issuance constraint ratio q (the parameters of the left-

and and right-hand side panels correspond to those of Points A and

in the upper left-hand side panel of Fig. 3, respectively). In the

eft-hand and right-hand side panels, the firm prefers market debt

ssuance and bank debt issuance, respectively. Recall from Lemma 2

hat the firm is constrained by the debt issuance bound if xpj

< xi∗jN

j ∈ {1, 2}), while it is not otherwise. In the upper left-hand side panel,

or example, if q < 0.9449 (q < 0.7045) because xp1 < xi∗

1N (xp2 < xi∗

2N),

he firm is constrained by the market (bank) debt issuance bound. We

ave limq↓0 xi∗j

= xi∗U and limq↑+∞ xi∗

j= xi∗

jNwith xi∗

U ≥ xi∗jN

. The right-

and side panel follows similarly.

Interestingly, xi∗j

is not always in between xi∗jN

and xi∗U for any q ∈

0, +∞), given a debt structure j ( j ∈ {1, 2}). This implies that the first





0 0.5 10

0.05

0.1

0.15

0.2

η (bargaining power)

σ (v

olat

ility

)

q=1q=0.8q=0.6

Eo1>Eo

2

Eo2>Eo

1

C

B

A

0 0.7045 0.9449

0.3057

0.4002

0.4349

q

Eo (

equi

ty o

ptio

n va

lue)

EUo

E1No

E1o

E2No

E2o

0.6 0.71 0.7945 0.92830.575

0.5928

0.6020

q

Eo (

equi

ty o

ptio

n va

lue)

E1No

E1o

E2No

E2o

0 0.9004 1.2

0.6698

0.7734

0.7869

0.8

q

Eo (

equi

ty o

ptio

n va

lue)

EUo

E1No

E1o

E2No

E2o

Fig. 3. Equity option values.

O

a

c

r

t

T

b

a

i

P

t

conjecture is no longer correct. That is, the investment thresholds xi∗j

are nonmonotonic with respect to q.

Consider the intuition of the U-shaped relationship between xi∗j

and q. When the firm comes up against the debt issuance limit con-

straints, an increase in q increases the debt value Daj(xi∗

j, c∗

j). Then there

are two conflicting effects of xi∗j

associated with increasing Daj(xi∗

j, c∗

j).

The first is the “financing constraint effect” where an increase in q

increases xi∗j

because Daj(xi∗

j, c∗

j) is increasing with respect to xi∗

j. The

second is the “leverage effect” where an increase in q decreases xi∗j

via

(x/xi∗j)β{Ea

j(xi∗

j, c∗

j)− (I − Da

j(xi∗

j, c∗

j))}. For low values of q, the second

effect dominates the first effect. For high values of q, the first effect

dominates the second effect. Thus, the investment thresholds have

a U-shaped relationship with respect to q.17 We summarize these

results as follows.

17 As demonstrated in Fig. 4, to the extent that we have solved the investment thresh-

olds numerically for various parameter values, we could not find any example of a

monotonic relationship between xi∗j

and q for any j.

a

s

t

l

t



bservation 3. Given a debt structure, the investment thresholds have

U-shaped relationship with the debt issuance bound q in almost all

ases.

Shibata and Nishihara (2012) have already shown a nonmonotonic

elationship between xi∗1 and q. The new result obtained in Observa-

ion 3 is that we find a nonmonotonic relationship between xi∗2 and q.

he nonmonotonicity property is consistent with theoretical studies

y Boyle and Guthrie (2003) and Hirth and Uhrig-Homburg (2010)

nd an empirical study by Cleary et al. (2007).

The two panels of Fig. 5 provide the most interesting case, assum-

ng that η = 1 and σ = 0.15 which correspond to the parameters of

oint B in the upper left-hand side of Fig. 3. Under these parame-

ers, we have already shown that the firm chooses bank debt issuance

t the time of investment for 0 < q < q̂ = 0.71, and market debt is-

uance for q ≥ q̂ = 0.71. The thick dotted line shows the investment

hresholds (coupon payments) at the equilibrium. An increase in q

eads to a jump in the investment thresholds, because the debt struc-

ure strategy has changed from bank debt to market debt. We see





0 0.7045 0.9449 1.2

0.5832

0.5962

0.6404

q

xi (in

vest

men

t thr

esho

ld)

xUi∗

x1Ni∗

x1i∗

x1p

x2Ni∗

x2i∗

x2p

0 0.9317 1.2

0.7572

0.7627

0.78

0.8101

q

xi (in

vest

men

t thr

esho

ld)

xUi∗

x1Ni∗

x1i∗

x1p

x2Ni∗

x2i∗

x2p

0 0.7045 0.9449 1.20

0.3401

0.4405

0.5

q

c (c

oupo

n pa

ymen

t)

c1N∗

c1∗

c2N∗

c2∗

0 0.9004 1.20

0.4623

0.5161

0.6

q

c (c

oupo

n pa

ymen

t)

c1N∗

c1∗

c2N∗

c2∗

Fig. 4. Investment thresholds and coupon payments with q.

t

t

O

l

t

p

t

c

n

4

t

r

t

s

O

c

t

i

t

M

t

a

18 To the extent that we have solved the coupon payments numerically for various

parameter values, we could not find any example of a nonmonotonic relationship

between c∗j

and q for any j ( j ∈ {1, 2}).

hat, at q = 0.71 the investment threshold jumps from xi∗2 = 0.6612

o xi∗1 = 0.6585. We have the following result.


evered firm defined by (22). Then, at the equilibrium, the investment

hresholds have a discontinuous W-shaped relationship with the friction

arameter q, depending on the two other key parameters, η and σ .

With the optimal choice of debt structure, the relationship be-

ween the investment thresholds and debt issuance bound is a more

omplicated nonmonotonicity relationship, compared with the sce-

ario in which firms have no choice of debt structure.

.2. Effects of constraints on coupon payments

This subsection examines the effects of a debt issuance bound on

he coupon payments.

The lower two panels of Fig. 4 show the coupon payments with

espect to q. We see that c∗j

is monotonically increasing with respect



o q, given the debt structure j ( j ∈ {1, 2}).18 The next observation

ummarizes this property of the coupon payments.

bservation 5. Given a debt structure, the coupon payments for the

onstrained levered firm are between those for the unlevered firm and

hose for the nonconstrained levered firm in almost all cases.

Observation 5 contrasts with Observation 3, where xi∗j

is not always

n the regions [xi∗jN

, xi∗U ]. Thus, it is less costly to distort xi∗

jaway from

he regions [xi∗jN

, xi∗U ] than to distort c∗

jaway from the regions [0, c∗

jN].

oreover, note that c∗j

≤ c∗jN

implies xd∗j

= xdj(c∗

j) ≤ xd

j(c∗

jN) = xd∗

jN.

In the right-hand side panel of Fig. 5, the thick dotted line shows

he coupon payments at the equilibrium. An increase in q leads to

jump in the coupon payments from c∗2 = 0.3510 to c∗

1 = 0.3302 at





0.5 0.63 0.71 0.7945 0.9283

0.6585

0.6612

0.6698

0.6725

q

xi (in

vest

men

t thr

esho

ld)

x1Ni∗

x1i∗

x2Ni∗

x2i∗

eq.

0.5 0.71 0.7945 0.9283

0.3302

0.3510

0.4146

0.4454

q

c (c

oupo

n pa

ymen

ts)

c1N∗

c1∗

c2N∗

c2∗

eq.

Fig. 5. Discontinuous solutions with respect to q.

0.1 0.1579 0.20.575

0.75

σ

x ji (in

vest

men

t thr

esho

ld)

xi∗1

xi∗2

0.1 0.1495 0.20.4

0.8

σ

Eo j (

equi

ty o

ptio

n va

lue)

Eo1

Eo2

Fig. 6. Asymmetric relationship between thresholds and values.

a

s

p

s

W

O

t

w

m

fi

19 For all the regions of σ ∈ [0.1, 0.2] with q = 0.7 and η = 1, we have xi∗jN

> xpj

( j ∈{1, 2}), implying that the firm is financially constrained by debt issuance bounds.

20 We confirm one of the results obtained in Fig. 2: namely, that an increase in σ

changes the firm’s debt capital strategies from market debt financing to bank debt

financing.

q = 0.71. The reason is that the firm changes the debt structure strate-

gies at q̂ = 0.71. We summarize the result as follows.

Observation 6. Consider the optimization problem for the constrained

levered firm defined by (22). Then, at the equilibrium, the coupon pay-

ments have a discontinuous curved relationship with the friction param-

eter q, depending on the two other key parameters, η and σ .

4.3. Relationship between investment thresholds and values

In this subsection, we consider the relationship between the in-

vestment thresholds and the corresponding values. Recall in Lemma 1

that we have a symmetric relationship (i.e., xi∗1N ≤ xi∗

2N if and only if

Eo1N(x) ≥ Eo

2N(x)) when q is sufficiently large. This result implies the

second intuitive conjecture, i.e., xi∗1 ≤ xi∗

2 if and only if Eo1(x) ≥ Eo

2(x).Fig. 6 shows the effects of volatility, σ , on the investment thresh-

olds and equity options values, respectively. The key parameters are



ssumed to be q = 0.7 and η = 1 for σ ∈ [0.1, 0.2].19 In the left-hand

ide panel, we have xi∗1 < xi∗

2 for σ < 0.1579. In the right-hand side

anel, we see that Eo1(x) ≥ Eo

2(x) for σ ≤ 0.1495.20 From these two re-

ults, we obtain xi∗1 < xi∗

2 and Eo1(x) < Eo

2(x) for σ ∈ (0.1495, 0.1579).e have the following result.

bservation 7. Under debt issuance limit constraints, a symmetric rela-

ionship is not always obtained. In particular, the investment thresholds

ith bank (market) debt financing are not always lower than those with

arket (bank) debt financing when the firm prefers bank (market) debt

nancing to market (bank) debt financing.





i

d

s

t

4

i

w

i

i

h

o

d

c

t

(

c

o

E

l

[

d

c

e

k

O

a

o

t

t

o

m

e

e

c

4

s

s

r

d

c

k

r

a

c

T

e

fi

b

d

t

r

s

o

P

fi

c

F

d

d

c

w

i

l

d

t

a

r

l

B

fi

O

d

d

T

c

p

i

fi

f

f

t

t

fi

fi

t

O

d

o

s

p

4

e

h

o

t

d

i

t

d

d

Note that the finding in Observation 7 provides important new

nsights into the difference between the model with the choice of

ebt structure (our model) and the model without the choice of debt

tructure (the model of Shibata and Nishihara (2012)), as shown in

he next subsection.

.4. Effects of having the additional choice of debt structure

This subsection examines how debt structure choice (market debt

ssuance or bank debt issuance) influences corporate investment

hen the firm is constrained with a debt issuance bound. Recall that

t is assumed that the firm can issue only market debt at the time of

nvestment in the previous model developed by Shibata and Nishi-

ara (2012). Thus, this subsection makes clear the difference between

ur model and the previous model.

Suppose, as a benchmark, that the firm is not constrained by a

ebt issuance bound (i.e., q is sufficiently large). Then, having the

hoice of market debt issuance or bank debt issuance always increases

he equity options values and always hastens corporate investment

decreases the investment threshold), compared with having only the

hoice of market debt issuance. The reason is a straightforward result

f the symmetric relationship (i.e., xi∗1N ≥ xi∗

2N if and only if Eo1N(x) ≤

o2N(x)).

Suppose that the firm comes up against the debt issuance

imit constraint (for example, assume q = 0.7, η = 1, and σ ∈0.1495, 0.1579]). Then, interestingly, having the choice of market

ebt issuance or bank debt issuance delays corporate investment (in-

reases the investment threshold) although it always increases the

quity options value, compared with having only the choice of mar-

et debt issuance. Thus, we summarize these results as follows.

bservation 8. Suppose that the firm faces the debt issuance constraint

t the time of investment. Then, having the choice of market debt issuance

r bank debt issuance does not always hasten corporate investment al-

hough it always increases the equity options value.

Our model is an extension of Shibata and Nishihara (2012) in that

he firm can choose the optimal debt structure (i.e., the firm has the

ption of issuing bank debt). Our intuition is that choosing the opti-

al debt structure enables the firm to hasten its corporate investment

ven though the firm is constrained by the debt issuance bound. How-

ver, interestingly, this intuition is not always correct when the firm

omes up against financial constraints.

.5. Risk and return for debt holders

In this subsection, we consider the effects of debt issuance con-

traints on debt holders. To do so, the two key variables are credit

preads and default probabilities, which are regarded as measures of

eturn and risk for debt holders, respectively. The credit spread and

efault probability are defined as

sk := c∗k

Daj(xi∗

k, c∗

k)

− r, pk := ET i∗k

[e−r(T i∗k

−Td∗k

)] =(

xi∗k

xdj(c∗

k)

)γ

,

∈ {j, jN}, (42)

espectively (j ∈ {1, 2}). Note that the two measures above are defined

t the optimal investment time T i∗k

.

We have already recognized that debt issuance constraints cause

∗j ≤ c∗

jN, xd∗j ≤ xd∗

jN , Daj

(xi∗

j , c∗j

) ≤ Daj

(xi∗

jN, c∗jN

),

j ∈ {1, 2}. (43)

he first inequality has already been shown in the lower two pan-

ls of Fig. 4; the second inequality is straightforward because of the

rst inequality and xd∗k

= xdj(c∗

k); and the third inequality is obvious



ecause of the definition of debt issuance constraints. Moreover, the

ebt issuance constraint does not always cause xi∗j

≥ xi∗jN

, as shown in

he upper two panels of Fig. 4. Thus, using these previously presented

esults, we cannot determine which is bigger: csj or csjN and pj or pjN.

The upper two and lower two panels of Fig. 7 depict the credit

preads and the default probabilities, respectively. The parameters

f the left-hand and right-hand side panels correspond to those of

oints A and C in the upper left-hand side panel of Fig. 3, where the

rm prefers market and bank debt financing, respectively. We see

sj ≤ csjN, pj ≤ pjN. (44)

or both market and bank debt financing, we have the result that the

ebt issuance limit constraint causes a decrease in credit spreads and

efault probabilities. Regarding the first inequality, the debt issuance

onstraint leads to a decrease in the ratio of c∗j

to Daj(xi∗

j, c∗

j), compared

ith the ratio of c∗jN

to DajN

(xi∗jN

, c∗jN

). The implication of the second

nequality is that, because γ < 0, the distance between xi∗j

and xd∗j

is

arger than the distance between xi∗jN

and xd∗jN

. This implies that the

efault for the constrained levered firm is less likely to occur than

hat for the unconstrained levered firm.

Thus, the more severe the debt issuance bounds are, the lower

re the credit spreads and default probabilities. These links between

isk and return are the same as those traditionally suggested in the

iterature and also match the results by Gomes and Schmid (2010) and

orgonovo and Gatti (2013). The next characterizes the properties of

nancing constraints.

bservation 9. Debt financing constraints lead to a low-risk (low-

efault probabilities) and low-return (low-credit spreads) situation for

ebt holders.

In addition, in both panels of Fig. 7, we have cs2 ≥ cs1 and p2 ≥ p1.

hese inequalities imply that bank debt is high risk and high return,

ompared with market debt. Importantly, irrespective of the firm’s

reference of debt structure, the credit spread and default probabil-

ties for bank debt financing are larger than those for market debt

nancing. Regarding the first inequality, the ratio of c∗2 to Da

2(xi∗2 , c∗

2)

or bank debt issuance is higher than the ratio of c∗1 to Da

1(xi∗1 , c∗

1)or market debt issuance. The implication of the second inequality is

hat, because of γ < 0, the distance between xi∗2 and xd∗

2 is smaller

han the distance between xi∗1 and xd∗

1 , implying that default of the

rm financed by bank debt is more likely to occur than default of the

rm financed by market debt. These numerical simulations suggest

he following observation.

bservation 10. The credit spreads and default probabilities for bank

ebt are larger than those for market debt in almost all cases.

The results presented are consistent with the empirical results

f Davydenko and Strebulaev (2007) who find that the possibility of

trategic debt service increases corporate credit spreads and default

robabilities.

.6. Amount of debt issuance

So far, we have assumed that the debt issuance amount (i.e., q) is

xogenous. This subsection examines how q is determined by debt

olders for a given debt structure j ( j ∈ {1, 2}).

Note that there is no arbitrage opportunity for debt issuance in

ur model, where debt holders provide the amount of qI and take

he value of Daj(xi∗

j, c∗

j) = qI. Additionally, credit spreads (return) and

efault probabilities (risk) are monotonically increasing with q. This

mplies that a decrease in q will enable debt holders to decrease

heir return and risk. In practice, it is natural for debt holders to

etermine q assuming, e.g., credit spreads are lower than 20 bp or

efault probabilities are lower than 5 percent. We now suppose that





0 0.57 0.65 0.85 1.20

20

32.5119

40

65.6535

q

cs (

cred

it sp

read

, bp)

cs1N

cs1

cs2N

cs2

0 0.33 0.46 0.63 0.90 1.20

20

40

92.3965

246.4458

q

cs (

cred

it sp

read

, bp)

cs1N

cs1

cs2N

cs2

0 0.48 0.55 0.90 1.20

56.0283

10

39.2431

q

p (d

efau

lt pr

obab

ility

, %)

p1N

p1

p2N

p2

0 0.30 0.42 0.56 0.79 0.90 1.20

5

10

13.9647

62.0389

70

q

p (d

efau

lt pr

obab

ility

, %)

p1N

p1

p2N

p2

Fig. 7. Low-risk and low-return of debt holders by financing constraints.

t

u

a

r

a

f

5

k

5

s

d

s

fi

a

c

t

q is determined on the basis of amount of return and risk debt holders

assume.

In the upper left-hand side panel of Fig. 7 (σ = 0.1), suppose

that q is decided such that credit spreads are lower than 20 bp. We

then have q = 0.5759 and q = 0.8564 for the bank debt and mar-

ket debt, respectively, which gives the result Da2(x

i∗2 , c∗

2) = 2.8795 and

Da1(x

i∗1 , c∗

1) = 4.2820. Additionally, suppose that credit spreads are al-

lowed up to 40 bp. We then have q = 0.6486 (i.e., Da2(x

i∗2 , c∗

2) = 3.2430)

and q = 0.9449 (i.e., Da1(x

i∗1 , c∗

1) = 4.7245). By increasing the upper

limit constraints of credit spreads, q is increased. In the upper right-

hand side panel (σ = 0.2), for the upper limit constraints of credit

spreads allowed up to 20 bp, we have q = 0.3385 and q = 0.4648 for

bank and market debt, respectively. Thus, an increase in σ decreases

q (i.e., the debt issuance amount).

Similarly, we examine the upper limit constraints of default prob-

abilities. In the lower left-hand side panel of Fig. 7 (σ = 0.1), sup-

pose that q is decided such that the default probability is lower than

5 percent. We then have q = 0.4856 (i.e., Da2(x

i∗2 , c∗

2) = 2.4280) and q =0.9077 (i.e., Da(xi∗, c∗) = 4.5385) for bank and market debt, respec-
1 1 1


ively. Additionally, suppose that default probabilities are allowed

p to 10 percent. We then have q = 0.5525 (i.e., Da2(x

i∗2 , c∗

2) = 2.7625)

nd q = 0.9449 (i.e., Da1(x

i∗1 , c∗

1) = 4.7245) for bank and market debt,

espectively. By increasing the upper limit constraints of default prob-

bilities, q is increased. In the lower right-hand side panel (σ = 0.2),

or the upper limit constraints of default probabilities allowed up to

percent, we then have q = 0.3031 and q = 0.5688 for bank and mar-

et debt, respectively. Thus, an increase in σ decreases q.

. Concluding remarks

We have proposed a model that analyzes optimal investment

trategies under an optimal structure of bank debt and market

ebt with issuance limit constraints. By considering various debt

tructures, we shed light on decisions regarding a firm’s investment,

nancing, and debt choice strategies. As a result, we can discuss the

vailability and effects of various debt structures under issuance

onstraints.

We have four important results with respect to financial fric-

ions. The first result is which type of debt is issued at the time of





i

v

d

fi

t

l

o

m

b

s

n

b

c

O

t

T

c

a

i

h

t

c

s

s

t

t

u

t

a

T

d

d

o

u

f

a

s

a

p

W

c

A

e

t

(

C

v

U

v

c

I

K

J

m

A

P

(

t

s

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w

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a

A

F

F

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a

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B

B

B

B

B

B

B

B

nvestment depends largely on three parameters (debt issuance limit,

olatility, and bargaining power in the renegotiation during financial

istress). In particular, increasing the debt issuance limit makes a

rm more likely to issue market debt than bank debt. In practice,

he firms with higher (lower) debt issuance bounds are regarded as

arge/mature (small/young) corporations. Based on this definition,

ur results show that large/mature (small/young) corporations are

ore likely to choose market (bank) debt. Thus, this segmentation is

roadly consistent with the stylized facts.

The second result is that, for a given debt structure, the relation-

hip between investment thresholds and the debt issuance limit is

onmonotonic. Because we find a nonmonotonic relationship under

ank debt financing in addition to market debt financing, we can

onclude that the nonmonotonic relationship is theoretically robust.

n the other hand, for a given debt structure, the relationship be-

ween coupon payments and the debt issuance limit is monotonic.

hus, it is less costly to distort investment thresholds than to distort

oupon payments. Moreover, if the optimal debt structure strategies

re changed from bank debt to market debt by increasing the debt

ssuance bound, the investment thresholds (and coupon payments)

ave a discontinuous relationship with the debt issuance bound. Thus,

he optimal debt structure makes the corporate investment (and

oupon payments) strategy more complicated, compared with the

cenario in which firms have no choice of debt structure.

The third result is that there is not always a symmetric relation-

hip between investment thresholds and equity option values when

here is a debt issuance constraint. More precisely, the investment

hresholds under bank debt financing are not always lower than those

nder market debt financing, even when the firm prefers bank debt

o market debt. This result contrasts with the result where there is

symmetric relationship when there is no debt issuance constraint.

hus, under a debt issuance constraint, having the choice of market

ebt or bank debt (having the additional choice of debt structure)

oes not always accelerate corporate investment although its equity

ption value is increased. We conclude that investment strategies

nder financing frictions are different from those under no financing

rictions.

The final result is that debt issuance constraints create low-risk

nd low-return situation for debt holders. More precisely, the more

evere the debt issuance limits are, the lower are the credit spreads

nd default probabilities. In addition, the credit spreads and default

robabilities for bank debt are larger than those for market debt.

e conclude that bank debt is relatively high risk and high return,

ompared with market debt.

cknowledgments

We thank seminar participants at the Advanced Methods in Math-

matical Finance Conference (Angers), the 7th World Congress of

he Bachelier Finance Society (Sydney), the EURO XXVI Conference

Rome), the 12th SAET Conference (Brisbane), the Bank of Japan, the

hinese University of Hong Kong, the Fields Institute, Hokkaido Uni-

ersity, the Hong Kong University of Science and Technology, Kyoto

niversity, Nanzan University, National University of Singapore, Uni-

ersity College London, and the University of Tokyo for their helpful

omments. This research was supported by a Grant-in-Aid from the

shii Memorial Securities Research Promotion Foundation, the JSPS

AKENHI (Grant Numbers 26242028, 26285071, and 26350424), the

SPS Postdoctoral Fellowships for Research Abroad, and the Telecom-

unications Advances Foundation.

ppendix A

In this subsection, we derive the functions fj1, fj2, fj3, and fj4 in

roposition 1. Note that fjk is a function of (xi∗j

, c∗j) for all j and k

j ∈ {1, 2}, k ∈ {1, 2, 3, 4}). Because the derivations of f and f are
1k 2k


he same (k ∈ {1, 2, 3, 4}), we show only the derivation of f2k. In the

ubgame to the investment decision problem for the firm with con-

traint bank debt financing, the Lagrangian is formulated as

= xi2

−β(�xi

2 + τ c2

r− τ c2

r

β

β − γ

(κ2xi

2

c2

)γ

− I

)

+λ

(qI − c2

r− c2

((1 − αη)

�

κ2

− 1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

))(κ2xi

2

c2

)γ )), (A.1)

here λ ≥ 0 denotes the multiplier on the constraint. The Karush–

uhn–Tucker conditions are given by

∂L∂xi

2

= −βxi∗2

−β−1(�xi

2 + τ c2

r− τ c2

r

β

β − γ

(κ2xi

2

c2

)γ

− I

)

+ xi∗2

−β(� − τ c∗

2

r

β

β − γ

(κ2xi∗

2

c∗2

)γ

γ xi∗2

−1)

−λc∗2

((1 − αη)

�

κ2− 1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

))

×(

κ2xi∗2

c∗2

)γ

γ xi∗2

−1 = 0, (A.2)

∂L∂c2

= xi∗2

−β(

τ

r− τ

r

β

β − γ

(κ2xi∗

2

c∗2

)γ

(1 − γ )

)

−λ

(1

r+

((1 − αη)

�

κ2− 1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

))

× (1 − γ )

(κ2xi∗

2

c∗2

)γ )= 0, (A.3)

nd

λ

(qI − c∗

2

r− c∗

2

((1 − αη)

�

κ2− 1

r

(1 − τ + τ

β

β − γ− τ

ηγ

β − γ

))

×(

κ2xi∗2

c∗2

)γ ))= 0. (A.4)

ssume that λ > 0, which is equivalent to xi∗2 > x1 from Lemma 2.

or q = 0, because c∗2 = 0 from (A.4), we have xi∗

2 = xi∗U and xd∗

2 = 0.

or q > 0, we have c∗2 > 0. Then (xi∗

2 , c∗2)are obtained uniquely by two

imultaneous equations:

f21(xi∗2 , c∗

2)

f22(xi∗2 , c∗

2)− f23(xi∗

2 , c∗2)

f24(xi∗2 , c∗

2)= 0, Da

2(xi∗2 , c∗

2)− qI = 0. (A.5)

he first equation is given by rearranging (A.2) and (A.3), while the

econd equation is given by (A.4). Thus, f21, f22, f23, and f24 are obtained

s (37)–(40). Similarly, we can derive f11, f12, f13, and f14.

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orgonovo, E., & Gatti, S. (2013). Risk analysis with contractual default. Does covenantbreach matter? European Journal of Operational Research, 230(2), 431–443.

oyle, G. W., & Guthrie, G. A. (2003). Investment, uncertainty, and liquidity. Journal ofFinance, 63(5), 2143–2166.

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Investment timing, debt structure, and financing constraints